1. Administrative Sep. 27 (today) – HW4 due Sep. 28 8am – problem session Oct. 2 Oct. 4 – QUIZ #2 (pages 45-79 of DPV)

2. Recap algorithm for k-select with O(n) worst-case running time modification of quick-sort which has O(n.log n) worst-case running time randomized k-select GOAL: O(n) expected running-time

3. Finding the k-th smallest element Select(k,A[c..d]) Split(A[c..d],x) xx xx j j  k  k-th smallest on left j<k  (k-j)-th smallest on right x=random element from A[c..d]

4. Finite probability space set  (sample space) function P:  R + (probability distribution) elements of  are called atomic events subsets of  are called events probability of an event A is  P(x) xAxA P(A)=  P(x) = 1 x 

5. Examples A B C Are A,B independent ? Are A,C independent ? Are B,C independent ? Is it true that P(A  B  C)=P(A)P(B)P(C)?

6. Examples A B C Are A,B independent ? Are A,C independent ? Are B,C independent ? Is it true that P(A  B  C)=P(A)P(B)P(C)? Events A,B,C are pairwise independent but not (fully) independent

7. Full independence Events A 1,…,A n are (fully) independent If for every subset S  [n]:={1,2,…,n} P (  A i ) =  P(A i ) iSiS iSiS

8. Random variable set  (sample space) function P:  R + (probability distribution)  P(x) = 1 x  A random variable is a function Y :  R The expected value of Y is E[X] :=  P(x)* Y(x) x 

9. Examples Roll two dice. Let S be their sum. If S=7 then player A gives player B $6 otherwise player B gives player A $1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

10. Examples Roll two dice. Let S be their sum. If S=7 then player A gives player B $6 otherwise player B gives player A $1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 -1, -1,-1,-1, -1, 6,-1,-1, -1, -1, -1 Expected income for B E[Y] = 6*(1/6)-1*(5/6)= 1/6 Y:

11. Linearity of expectation E[X  Y]  E[X] + E[Y] E[X 1  X 2  …  X n ]  E[X 1 ] + E[X 2 ]+…+E[X n ] LEMMA: More generally:

12. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. Let n be the number of people in the class. For what n is the game advantageous for me?

13. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise E[X 1 ] = ?

14. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise E[X 1 ] = -9/n + 1*(n-1)/n

15. Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise X 2 = -9 if player 2 gets his card back 1 otherwise E[X 1 +…+X n ] = E[X 1 ]+…+E[X n ] = n ( -9/n + 1*(n-1)/n ) = n – 10.

16. Do you expect to see the expected value? X= 1 with probability ½ 3 with probability ½ E[X] =

17. Expected number of coin-tosses until HEADS? H ½ TH ¼ TTH 1/8 TTTH 1/16 TTTTH 1/32....

18. Expected number of coin-tosses until HEADS?   n.2 -n = 2 n=1  Expected number of dice-throws until you get “6” ?

19. Finding the k-th smallest element Select(k,A[c..d]) Split(A[c..d],x) xx xx j j  k  k-th smallest on left j<k  (k-j)-th smallest on right x=random element from A[c..d]

20. FFT

21. Polynomials p(x) = a 0 + a 1 x +... + a d x d Polynomial of degree d

22. Multiplying polynomials p(x) = a 0 + a 1 x +... + a d x d Polynomial of degree d q(x) = b 0 + b 1 x +... + b d’ x d’ Polynomial of degree d’ p(x)q(x) = (a 0 b 0 ) + (a 0 b 1 + a 1 b 0 ) x +.... + (a d b d’ ) x d+d’

23. Polynomials p(x) = a 0 + a 1 x +... + a d x d THEOREM: A non-zero polynomial of degree d has at most d roots. Polynomial of degree d COROLLARY: A polynomial of degree d is determined by its value on d+1 points.

24. COROLLARY: A polynomial of degree d is determined by its value on d+1 points. Find a polynomial p of degree d such that p(a 0 ) = 1 p(a 1 ) = 0.... p(a d ) = 0

25. COROLLARY: A polynomial of degree d is determined by its value on d+1 points. Find a polynomial p of degree d such that p(a 0 ) = 1 p(a 1 ) = 0.... p(a d ) = 0 (x-a 1 )(x-a 2 )...(x-a d ) (a 0 -a 1 )(a 0 -a 2 )...(a 0 -a d )

26. Representing polynomial of degree d d+1 coefficients evaluation on d+1 points the coefficient representation the value representation evaluationinterpolation

27. Evaluation on multiple points p(x) = 7 + x + 5x 2 + 3x 3 + 6x 4 + 2x 5 p(z) = 7 + z + 5z 2 + 3z 3 + 6z 4 + 2z 5 p(-z) = 7 – z + 5z 2 – 3z 3 + 6z 4 – 2z 5 p(x) = (7+5x 2 + 6x 4 ) + x(1+3x 2 + 2x 4 ) p(x) = p e (x 2 ) + x p o (x 2 ) p(-x) = p e (x 2 ) – x p o (x 2 )

28. Evaluation on multiple points p(x) = a 0 + a 1 x + a 2 x 2 +... + a d x d p(x) = p e (x 2 ) + x p o (x 2 ) p(-x) = p e (x 2 ) – x p o (x 2 ) To evaluate p(x) on -x 1,x 1,-x 2,x 2,...,-x n,x n we only evaluate p e (x) and p o (x) on x 1 2,...,x n 2

29. Evaluation on multiple points To evaluate p(x) on -x 1,x 1,-x 2,x 2,...,-x n,x n we only evaluate p e (x) and p o (x) on x 1 2,...,x n 2 To evaluate p e (x) on x 1 2,...,x n 2 we only evaluate p e (x) on ?

30. n-th roots of unity 2  ik/n e  k  n = 1  k.  l =  k+l  0 +  1 +... +  n-1 = 0 FACT 1: FACT 2: FACT 3: FACT 4:  k = -  k+n/2

31. FFT (a 0,a 1,...,a n-1,  ) (s 0,...,s n/2-1 )= FFT(a 0,a 2,...,a n-2,  2 ) (z 0,...,z n/2-1 ) = FFT(a 1,a 3,...,a n-1,  2 ) s 0 + z 0 s 1 +  z 1 s 2 +  2 z 2.... s 0 – z 0 s 1 -  z 1 s 2 -  2 z 2....

32. Evaluation of a polynomial viewed as vector mutiplication (a 0,a 1,a 2,...,a d ) 1xx2..xd1xx2..xd

33. Evaluation of a polynomial on multiple points (a 0,a 1,a 2,...,a d ) 1x1x12..x1d1x1x12..x1d 1x2x22..x2d1x2x22..x2d 1xnxn2..xnd1xnxn2..xnd... Vandermonde matrix